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A062734
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Triangular array T(n,k) giving number of connected graphs with n labeled nodes and k edges (n >= 1, 0 <= k <= n(n-1)/2).
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17
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1, 0, 1, 0, 0, 3, 1, 0, 0, 0, 16, 15, 6, 1, 0, 0, 0, 0, 125, 222, 205, 120, 45, 10, 1, 0, 0, 0, 0, 0, 1296, 3660, 5700, 6165, 4945, 2997, 1365, 455, 105, 15, 1, 0, 0, 0, 0, 0, 0, 16807, 68295, 156555, 258125, 331506, 343140, 290745, 202755, 116175, 54257, 20349
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OFFSET
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1,6
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COMMENTS
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T(n,n-1) = n^(n-2) counts free labeled trees A000272.
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REFERENCES
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Cowan, D. D.; Mullin, R. C.; Stanton, R. G. Counting algorithms for connected labelled graphs. Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), pp. 225-236. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Man., 1975. MR0414417 (54 #2519). - N. J. A. Sloane, Apr 06 2012
F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, Page 29, Exercise 1.5.
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LINKS
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FORMULA
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G.f.: Sum_{n>=1, k>=0} T(n, k) * x^n/n! * y^k = log(Sum_{n>=0} (1 + y)^binomial(n, 2) * x^n/n!). - Ralf Stephan, Jan 18 2005
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EXAMPLE
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Triangle starts:
[1],
[0, 1],
[0, 0, 3, 1],
[0, 0, 0, 16, 15, 6, 1],
[0, 0, 0, 0, 125, 222, 205, 120, 45, 10, 1],
...
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MATHEMATICA
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nn=6; s=Sum[(1+y)^Binomial[n, 2] x^n/n!, {n, 0, nn}]; Range[0, nn]!CoefficientList[Series[Log[ s]+1, {x, 0, nn}], {x, y}]//Grid (* returns triangle indexed at n = 0, Geoffrey Critzer, Oct 07 2012 *)
T[ n_, k_] := If[ n < 0, 0, Coefficient[ n! SeriesCoefficient[ Log[ Sum[ (1 + y)^Binomial[m, 2] x^m/m!, {m, 0, n}]], {x, 0, n}], y, k]]; (* Michael Somos, Aug 12 2017 *)
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PROG
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(PARI) {T(n, k) = if( n<0, 0, n! * polcoeff( polcoeff( log( sum(m=0, n, (1 + y)^(m * (m-1)/2) * x^m/m!)), n), k))}; /* Michael Somos, Aug 12 2017 */
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CROSSREFS
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See A123527 for another version (without leading zeros in each row).
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KEYWORD
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easy,nonn,tabf
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AUTHOR
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STATUS
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approved
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